It is well known that binomial identities or congruences usually have nice \(q\) -analogues (see [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}).
John Green [21]} proved a conjecture made by Daan Krammer. The identity was served as an inspiration for searching \(q\) -analogues of some identities in [22]}, [23]}.
In particular, Tauraso [4]} proved the following generalization of (REF ):
\(\sum _{k=0}^{n-1}q^k {\genfrac[]{0.0pt}{}{2k}{k+d}}_{}\equiv \left(\frac{n-|d|}{3}\right)q^{3r(r+1)/2+|d|(2r+1)}\ ({\rm {mod}}\ \Phi _n(q))\) 
